ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

134 7. Tripartite entanglement in three-mode Gaussian states
residual Gaussian contangle cannot detect tripartite PPT entangled states. For example,
the residual Gaussian contangle in three-mode biseparable Gaussian states
(Class 4 of Ref. [94]) is always zero, because those bound entangled states are separable
with respect to all 1 × 2 bipartitions of the modes. In this sense we can
correctly regard the residual Gaussian contangle as an estimator of distillable tripartite
entanglement, being strictly nonzero only on fully inseparable three-mode
Gaussian states (Class 1 in the classification of Sec. 7.1.1).
7.3. Sharing structure of tripartite entanglement: promiscuous Gaussian
states
We are now in the position to analyze the sharing structure of CV entanglement in
three-mode Gaussian states by taking the residual Gaussian contangle as a measure
of tripartite entanglement, in analogy with the study done for three qubits [72] using
the residual tangle [59] (see Sec. 1.4.3).
The first task we face is that of identifying the three-mode analogues of the two
inequivalent classes of fully inseparable three-qubit states, the GHZ state [100],
Eq. (1.51), and the W state [72], Eq. (1.52). These states are both pure and
fully symmetric, i.e. invariant under the exchange of any two qubits. On the one
hand, the GHZ state possesses maximal tripartite entanglement, quantified by the
residual tangle [59, 72], with zero couplewise entanglement in any reduced state of
two qubits reductions. Therefore its entanglement is very fragile against the loss of
one or more subsystems. On the other hand, the W state contains the maximal twoparty
entanglement in any reduced state of two qubits [72] and is thus maximally
robust against decoherence, while its tripartite residual tangle vanishes.
7.3.1. CV finite-squeezing GHZ/W states
To define the CV counterparts of the three-qubit states |ψGHZ〉 and |ψW 〉, one must
start from the fully symmetric (generally mixed) three-mode CM σs of the form
σ α 3, Eq. (2.60). Surprisingly enough, in symmetric three-mode Gaussian states,
if one aims at maximizing, at given single-mode mixedness a ≡ √ Det α, either
the bipartite entanglement G i|j
τ in any two-mode reduced state (i.e. aiming at the
CV W -like state), or the genuine tripartite entanglement G res
τ
(i.e. aiming at the
CV GHZ-like state), one finds the same, unique family of states. They are exactly
the pure, fully symmetric three-mode Gaussian states (three-mode squeezed states)
with CM σ p s of the form σ α 3, Eq. (2.60), with α = a2, ε = diag{e + , e − } and
e ± = a2 − 1 ± (a 2 − 1) (9a 2 − 1)
4a
, (7.39)
where we have used Eq. (5.13) ensuring the global purity of the state. In general,
we have studied the entanglement scaling in fully symmetric (pure) N-mode
Gaussian states by means of the unitary localization in Sec. 5.2. It is in order to
mention that these states were previously known in the literature as CV “GHZtype”
states [236, 240], as in the limit of infinite squeezing (a → ∞), they approach
the proper (unnormalizable) continuous-variable GHZ state dx|x, x, x〉, a simultaneous
eigenstate of total momentum ˆp1 + ˆp2 + ˆp3 and of all relative positions
ˆqi − ˆqj (i, j = 1, 2, 3), with zero eigenvalues [239].